The propagation of internal Kelvin waves on the equatorial thermocline plays an important role in equatorial dynamics. Figure 1 [Tropical Atmosphere–Ocean (TAO) array data] shows that the equatorial ocean can be well-described within the framework of a two-layer model. The Coriolis parameter vanishes at the equator so that the interface between the shallow and the deep layer serves as a waveguide for various trapped waves, including eastward travelling Kelvin waves and westward travelling Rossby waves. Trapped gravity–inertial waves (or Poincaré waves) can propagate in both directions, although the first gravity–inertial wave mode has properties of a Rossby wave when it travels westward [mixed Rossby–gravity or Yanai wave, Gill (1982); see Fig. 6 below].
Boyd (1980) and Ripa (1982) considered nonlinear equatorial Kelvin waves and showed that they could steepen and overturn or break. A wave of depression deepens the thermocline and breaks on the forward face of the wave, while a wave of elevation raises the thermocline and tends to break on the rearward face. A broken wave of depression can form a jump [also called shocks or fronts: Lighthill (1978), Philander (1990)]. In this paper we propose a model for such fronts (we call them equatorial Kelvin fronts) and study the properties. Earlier, Fedorov and Melville (1995, 1996) and Fedorov (1997) studied properties of nonlinear Kelvin waves and fronts trapped near coastal boundaries; this work considers analogous phenomena in the equatorial ocean. Whether Kelvin waves are linear waves or nonlinear fronts determines their main characteristics, including the speed of wave propagation, dissipation rates, and the meridional structure of the flow.
Extensive data on Kelvin waves have been obtained recently, motivated in part by possible connection between the initial stages of El Niño and equatorial Kelvin waves which may precede this event: Any relaxation or reversal of the steady trade winds (the easterlies) results in the excitation of a Kelvin wave, which can affect El Niño (e.g., Federov 2000, Manuscript in preparation). The periods of equatorial Kelvin waves vary from two weeks (Philander 1984) to two months (Kessler et al. 1995) and even to intraseasonal timescales [up to four months: Johnson and McPhaden (1993)]. The motion is characterized by zero flux across the equator. The typical phase speed of the waves ranges from 2 to 3 m s−1 and depends upon the depth of the thermocline and nonlinear effects.
Are nonlinear effects important for equatorial Kelvin waves? Delcroix et al. (1991) studied equatorial Kelvin waves in the Pacific ocean through Geosat measurements of sea level and surface current anomalies. The measured phase speed of the waves was 2.82 ± 0.96 m s−1, while theoretical predictions of the linear phase speed were in the range 2.26 ± 1.02 m s−1. Kessler et al. (1995) looked at the displacement of the thermocline during passages of Kelvin waves and found amplitudes a = ±20 m, with the depth of the thermocline varying from 80 to 170 m. Eriksen et al. (1983) measured sea-level fluctuations of about 10 cm, which they attributed to signatures of internal waves. The sea level anomalies in the study by Delcroix et al. 1991 were 10–15 cm. Such strong anomalies can be associated with the passage of internal waves with amplitudes of some tens of meters.
The above data suggest that the typical nonlinearity (defined as the ratio of the wave amplitude to the thermocline depth or, alternatively, as the ratio of the phase speed correction to the linear phase speed) may range from about 10% to 30%. For weaker waves, the nonlinearity may increase several-fold due to the shoaling of the thermocline (Long and Chang 1990; Yang and Yu 1992), as it decreases from 180 to 40 m, and sometimes even to 20 m, going from west to east.
One might assume that the phase speed corrections can be attributed to the mean currents, and especially to the effect of the equatorial undercurrent. However, calculations by Johnson and McPhaden (1993) for linear Kelvin waves show that adding the mean currents corrects the phase speed by only 4%. Thus there is sufficient evidence that nonlinearity may be important for equatorial Kelvin waves.
Since Kelvin waves are observed in the Pacific as well as the Atlantic Ocean (Katz 1987), the present work is applicable for both basins. However, as the Pacific basin is of greater size, the waves there have longer time for nonlinear evolution. The zonal extent of the Pacific in the equatorial region is approximately 145°, or about 16 000 km. In the next sections we will show that this may give the wave enough time to break and evolve into a fully developed front. Our results may be applicable also to the atmosphere where equatorially trapped Kelvin waves are well documented. For a review see Gill (1982).
Solving the full nonlinear shallow-water equations numerically we demonstrate that the wave steepens and approaches breaking in a manner qualitatively similar to that discussed by Boyd (1980) and Ripa (1982). However, after breaking and some adjustment, the Kelvin front1 is formed with properties rather different from classical two-dimensional hydraulic jumps. Note that both Boyd and Ripa assumed geostrophic balance for the zonal component of the velocity. This assumption renders the nonlinear Kelvin waves straight-crested. Further, it permits the reduction of the problem of the wave evolution to a two-dimensional problem (with one spatial coordinate along the equator, and time). Thus, wave breaking and fronts in their formulations are essentially two-dimensional.
In our work we show that, although the geostrophic balance holds during the initial stages of the wave evolution, a full three-dimensional model is needed to describe the developed front. In such a model, the temporal derivative of the transverse or meridional velocity is retained in the y-momentum equation, permitting curved fronts.
The Kelvin front is convex to the east with the equator as the axis of symmetry. A packet of gravity–inertial (or Poincaré) waves is generated, which travels at the same speed as the front. Although the cross-equatorial flow measured at the equator remains zero, the propagation of a Kelvin front leads to a nonzero net mass transport that is directed away from the equator above the thermocline. The maximum of this transport occurs at approximately one Rossby radius off the equator.
We explain these and other properties of Kelvin fronts within a semianalytical theory based on jump conditions and the approximation of a gravity–inertial boundary layer behind the jump. (For instance, the asymptotic angle of the front is given by a simple function of the front amplitude.) The major differences between the numerics and the theory is that in the latter we postulate the existence of the jump, and, thereby, obtain the wave field behind the front, while in the former the jump emerges after the evolution of the originally smooth initial disturbance.
Finally, by introducing a sloping thermocline and moderate asymmetry into the problem, we demonstrate that the formation of the Kelvin front is a robust feature of the model. (Although Long and Chang 1990 and Yang and Yu 1992 have considered the propagation of Kelvin waves on a slowly varying thermocline, neither of the studies dealt with front formation.)
2. Formulation of the problem
Note that the typical breaking commonly associated with wave overturning never happens in our model since we have added eddy viscosity into the system. Instead, strong velocity gradients emerge in front of the wave. In the field this would lead to strong mixing and turbulence in the transitional region. We still refer to this phenomenon as wave breaking.
3. Nonlinear evolution of equatorial Kelvin waves and the formation of a Kelvin front
For the numerical runs presented in this section we take Δx = 0.008, and A = 0.17, B = 0.34, and C = 0.58. Other choices of A, B, and C are possible, but these were found to be optimal within the range of parameters considered. This particular choice of A and Δx gives ν = 0.0014, which corresponds to a dimensional value of the horizontal eddy diffusion of 103 m2 s−1 if we use c = 2.2 m s−1 for the linear phase speed of Kelvin waves. Increasing Δx, while fixing A, B, and C, changes the solutions only slightly, amplifying viscosity and widening the region associated with the hydraulic jump. For further details of the numerics see Fedorov and Melville (1996).
In Fig. 2 we present an example of Kelvin wave evolution. Starting from its initial Gaussian shape, the equatorially trapped disturbance steepens, which is revealed in the higher concentration of contour lines on the front (Figs. 2a,b). The wave approaches breaking, and a curved front begins to form (Fig. 2b). Finally, the wave evolves into a fully developed Kelvin front (Fig. 2c). The front is convex to the east, and a complex crest pattern has emerged behind it. In the next section we will show that this pattern consists of Poincaré waves.
Figure 3 presents the amplitude of the wave as a function of time. The amplitude is defined as the maximum displacement of the thermocline on the equator. Although the values of the amplitude vary only slightly, one can distinguish several important regions of the graph. Clear oscillations of the amplitude on the left-hand side of the plot are related to the initial nonlinear adjustment of the wave. Since our initial conditions (2.25)–(2.27) are not an exact solution of the shallow-water equations, a relatively weak Rossby wave must emerge and travel westward, carrying excess energy. [This is analogous to the linear problem: see Gill (1982), Philander (1981), and Philander et al. (1984).] The region where the amplitude varies very little corresponds to wave steepening close to breaking. The right-hand side of the graph, with a relatively steep slope, is related to the formation of a Kelvin front. As soon as the front is developed, it causes a higher dissipation rate of the wave field.
4. Resonant generation of Poincaré waves
The detailed structure of the front and the wave field behind it is shown in Fig. 4, while the structure of the transverse (cross-equatorial) velocity field is given in Fig. 5. One of the most striking features of the wave development of the front (Figs. 2c, 4, and 5) is the appearance of relatively short waves behind it. Identification of these waves is facilitated by considering their cross-equatorial velocity field since the typical asymmetric cell structure in Fig. 5 can correspond only to the first Poincaré (gravity–inertial) mode. The group speed of the very short Rossby waves is too small to keep pace with a fast Kelvin wave (Fig. 6), while the higher Poincaré modes would have more nodes in the y direction. The remaining Yanai (mixed Rossby–gravity) wave would have a nonzero transverse velocity at the equator.
The Poincaré waves are forced by the Kelvin front as a result of a direct resonance: The front is supercritical and moves slightly faster than a linear Kelvin wave so that it matches the phase speed of the Poincaré waves of an appropriate frequency (or wavelength) as shown in Fig. 6. This is similar to the case considered by Melville et al. (1989) and Tomasson and Melville (1992) in which a nonlinear coastal Kelvin wave generated secondary Poincaré waves in a strait. We emphasize, however, that in Melville et al. the Poincaré wave generation was a time-dependent process, leading the continual excitation of Poincaré waves and to decay of the Kelvin wave. In the present study the Poincaré wave pattern is attached to the front, travels with the same speed as the front, and remains almost steady with respect to the front after some initial development. This will be used in section 6 in constructing a semianalytical theory of steady equatorial Kelvin fronts.
Nevertheless, in the initial value problem the Kelvin front is only quasi-steady since in a finite time the effect of the jump can be felt only at a finite distance away from it. The Poincaré waves are slowly spreading behind the front, which can happen because the group velocity of Poincaré waves is slightly smaller than that of a nonlinear Kelvin wave (Fig. 6).
The resonant approach gives some insight into the possible influence of the mean currents. Their main effect will be to modify the resonant condition locally. This would result in a change in the wave field, which may possibly alter the local shape of the jump (see section 6). However, the qualitative behavior and properties of the Kelvin front should not change.
5. Formation of Kelvin fronts: Sloping thermocline and asymmetry
These equations are solved numerically as in the previous case. Figure 7 shows the evolution of the initial disturbance advancing on the sloping thermocline. The concentration of the isolines in the frontal part of the wave is now due to two factors: the reduction of the local speed of the wave because of the decreasing depth of the thermocline and, second, the effect of nonlinearity itself. Both factors work to cause the wave to break. Figure 8 gives the evolution of the absolute wave amplitude that grows until the wave is broken and the front is formed. There are three distinct regions of behavior, as in the previous case (cf. Fig. 3). However, the initial relative amplitude is now smaller, as compared to the case of the flat thermocline, but still sufficient for the front to form. (The initial relative amplitude, defined as the ratio of the wave amplitude to the local depth of the thermocline, is now about 0.15.)
As we mentioned already, the propagation of the Kelvin waves on a slowly varying thermocline has been studied before. Yang and Yu (1992) used a linear model and apply the WKBJ method to obtain their solution. Long and Chang (1990) derived a KdV–type equation with varying coefficients for describing nonlinear Kelvin waves. Neither of the studies dealt with the fronts (hydraulic jumps). Importantly, in all previous studies it has been assumed that all different types of the equatorial waves become well separated after some time due to the differences in the phase speeds. As we have shown, it is not the case for the Kelvin jump for which Poincaré waves and the Kelvin wave do not separate. This is similar to the nonlinear Rossby adjustment in a channel (Tomasson and Melville 1992). The characteristic time scale of the nonlinear evolution is comparable to the time needed for the wave separation, in consequence of which full separation does not occur.
There is only one major difference in the solutions compared to the previous cases. An asymmetric Yanai wave emerges from the initial disturbance and follows the Kelvin front at a slower speed (Fig. 9). After some time, the front and Yanai wave become completely separated. After the separation the front remains symmetric and does not differ from the previous cases (Fig. 9c).
6. Three-dimensional fine structure of Kelvin fronts
From the previous examples, we conclude that for a range of the initial conditions and thermocline structure an equatorial disturbance evolves into a Kelvin front, provided the disturbance has a sufficient amplitude. The front propagates with little change in shape or speed. It is possible to construct a semi-analytical theory for such a steady front, based on the classical hydraulic theory and a boundary layer approach. In this model the hydraulic jump is moving eastward with a constant speed and is trailed by a wave wake. Mass and momentum are conserved across the jump, which has negligible width. The analysis below closely follows the derivations for the case of coastal hydraulic jumps with rotation (Fedorov and Melville 1996) and is given here only briefly.
We take the inviscid shallow-water equations (2.14)–(2.17) and their flux-conserving counterparts (2.18)–(2.20) as the basis for our analysis. We transform to a frame of reference travelling eastward with the front. The speed of the front is 1 + s, where s = Δc/c = O(α). That is, the speed of the jump is the phase speed of a linear Kelvin wave plus a nonlinear fractional correction s.
We solve the system (6.11)–(6.17) numerically by introducing time-dependence in the equations and pursuing calculations until a steady limit is reached. The details of the numerical approach are analogous to those given in Fedorov and Melville (1996).
Figures 10 and 11 display a Kelvin front, the displacement of the thermocline, and the transverse velocity field behind the jump, which is obtained by solving Eqs. (6.11)–(6.16), while Fig. 12 shows a full view of the Kelvin jump. Note that there is a difference between the steady jump solution in this section and the solution of an initial value problem in section 3 since the latter is only quasi-steady and its wave field goes to zero for large x. In spite of this, the two solutions in Fig. 10 (11) and 4 (5) are very similar, especially in the vicinity of the jump.
Figures 13 and 14 show the displacement of the thermocline and the transverse velocity field as a function of x at different latitudes. There is a striking difference when compared with two-dimensional jumps. For instance, one can clearly see Poincaré waves, slowly decaying with x. Note that, since the wave field decays only slowly, the boundary layer extends far upstream and, in fact, becomes a lengthy boundary region behind the jump. In the boundary region, the characteristic scale O(
Finally, Fig. 15 displays a cross section of the displacement of the thermocline along a meridian at some distance behind the most foremost point of the jump. There are overshoots caused by Poincaré waves at distances of 3.5Ro (approximately 1000 km) away from the equator, compared to a linear Kelvin wave solution. The overshoots are a result of the nonlinear dynamics. Delcroix et al. 1991 reported similar differences between the observed Kelvin waves and the linear theory occurring at the same distance, although the overshoots they detected were about two to three times as strong.
7. Off-equatorial mass transport
That is, the net meridional flow is nonzero and is proportional to the jump amplitude. The plot of T(y) in Fig. 16 shows two extrema in the net off-equatorial transport at distances approximately equal to the Rossby radius. In these internal waves, any mass transport is associated with a heat flux. In the waves of depression we have considered, warmer water masses are replacing colder waters. This implies that the off-equatorial mass transport (as in the Fig. 16) leads a proportional positive heat flux directed away from the equator.
We have demonstrated the possibility of Kelvin fronts on the equatorial thermocline. Whether or not the fronts appear depends upon the wavelength and amplitude of the initial disturbance, as well as the width of the basin. The shoaling of the thermocline facilitates the formation of the front. The front can emerge only from a wave of depression. Waves of elevation break on the rearward face, and are not considered here.
The properties of the Kelvin front derived from both the numerical model and semianalytical approach can be summarize as follows.
In the lee of the jump the wave field decays away from the equator in a quasi-Gaussian manner, which is similar to a regular Kelvin wave. Nevertheless, at some distance away from the jump there can be overshoots that distinguish the shape from pure Gaussian.
In the absence of dissipation the jump travels with a constant speed and maintains a permanent shape, which depends only on the jump strength. The dissipation and the limited extent of the initial conditions render the jump quasi-steady.
The jump curves back from the normal to the equator to straight oblique lines on each side of the equator. The angle included between these two lines and the meridian is a simple function of the jump amplitude at the equator.
As a result of resonant interactions Poincaré (i.e., trapped gravity–inertial) waves are generated behind the front and move with the same phase speed as the front. The resonance is possible because the speed of the front is slightly greater than that of a linear Kelvin wave. Together, the front and the Poincaré waves constitute a unified wave pattern. In contrast to the previous studies of the equatorial wave dynamics (e.g., Boyd 1980; Ripa 1982), in our model the nonlinear Kelvin and Poincaré waves do not separate.
Asymmetry of the initial conditions results in the generation of a Yanai (i.e., mixed Rossby–gravity) wave, which follows the front. However, after a short time the front and the Yanai wave become completely separated.
The Kelvin jump gives rise to a moderate net off-equatorial flow. Consequently, there is a contribution to the poleward heat flux. This feature is different from that of linear Kelvin waves, which have zero transverse flow.
There is indirect observational evidence that Kelvin fronts may exist. The amplitude of the observed Kelvin waves may be sufficient for the waves to break and for the fronts to emerge (e.g., Kessler et al. 1995). The observed meridional structure of the Kelvin waves is more complicated than a simple Gaussian distribution (Delcroix et al. 1991), analogous to the case considered here. There is some evidence of rapid temperature changes at the mooring sites of the TAO array (see, TAO Web site)3 and in the TOPEX/Poseidon data (J. Picaut 1998, personal communication), which show that the wind-forced Kelvin waves, sometimes associated with the El Niño signal in the eastern Pacific, are clearly fronts rather than linear Kelvin waves. This may necessitate some corrections of the Kelvin wave speed and dissipation rates used in current models of the ENSO. Another important consequence of the study is that nonlinear Kelvin waves may be a source for gravity–inertial waves on the equatorial thermocline. Also, the effect of Kelvin fronts on the mixing processes should be considered.
Finally, Kelvin fronts may also exist in the equatorial regions of the atmosphere where they would be unimpeded by coastal boundaries. This is a subject for further study.
AVF thanks Larry Pratt, Audrey Rogerson, Karl Helfrich, Joel Picaut, and Professor George Philander for helpful discussions. This work was supported by grants to WKM from the Office of Naval Research and National Science Foundation, and completed when AVF was a visiting research scientist at the AOS Program, Princeton University.
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Sometimes any nonlinear Kelvin wave is referred to as a Kelvin wave front. We choose to use the word front in the narrower sense:a wave led by a hydraulic jump or shock.
See Fedorov and Melville (1996) for a discussion.
Online at http://www.pmel.noaa.gov/toga-tao.